Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras

Abstract

Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the defining relations for $U_q(\mathfrak{g})$ can be slightly modified in such a way that the resulting algebra $U_q(\mathfrak{g};\mathcal{S})$ allows a homomorphism onto (an extension of) the algebra $\mathrm{Pol}({\mathbb{G}}_q/{\mathbb{K}}_{\mathcal{S},q})$ of functions on the quantum flag manifold ${\mathbb{G}}_q/{\mathbb{K}}_{\mathcal{S},q}$ corresponding to $\mathcal{S}$. Moreover, this homomorphism is equivariant with respect to a natural adjoint action of $U_q(\mathfrak{g})$ on $U_q(\mathfrak{g};\mathcal{S})$ and the standard action of $U_q(\mathfrak{g})$ on $Pol({\mathbb{G}}_q/{\mathbb{K}}_{\mathcal{S},q})$.

19 pages

References in corpus (2)

• A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions
• Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Cited by in corpus (2)

• Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices
• I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type